4

Attempts to Rethink Logic

JEREMY HEIS

To appear in The Cambridge History of Philosophy in the Nineteenth Century (1790-1870) (Cambridge, 2011)

The period between Kant and Frege is widely held to be an inactive time in the history of

logic, especially when compared to the periods that preceded and succeeded it. By the

late eighteenth century, the rich and suggestive exploratory work of Leibniz had led to

writings in symbolic logic by Lambert and Ploucquet.1 But after Lambert this tradition

effectively ended, and some of its innovations had to be rediscovered independently later

in the century. Venn characterized the period between Lambert and Boole as “almost a

blank in the history of the subject” and confessed an “uneasy suspicion” that a chief

cause was the “disastrous effect on logical method” wrought by Kant’s philosophy.2 De

Morgan began his work in symbolic logic “facing Kant’s assertion that logic neither has

improved since the time of Aristotle, nor of its own nature can improve.”3

1 J. H. Lambert, Sechs Versuche einer Zeichenkunst in der Vernunftslehre, in Logische und philosophische Abhandlungen, vol. 1, ed. J. Bernoulli (Berlin, 1782). G. Ploucquet, Sammlung der Schriften welche den logischen Calcul Herrn Professor Plocquet’s betreffen, mit neuen Zusätzen (Frankfurt, 1766).

2 John Venn, Symbolic Logic, 2nd ed. (London, 1894), xxxvii, original edition, 1881.3 Augustus De Morgan, “On the Syllogism III,” reprinted in On the Syllogism and Other

Writings, ed. P. Heath (New Haven, Conn.: Yale University Press, 1966), 74, original edition, 1858. De Morgan is referring to Immanuel Kant, Critique of Pure Reason, trans. Paul Guyer and Allen Wood (Cambridge: Cambridge University Press, 1998), Bviii. For the Critique of Pure Reason, I follow the common practice of citing the original page numbers in the first (A) or second (B) edition of 1781 and 1787. Citations of works of Kant besides the Critique of Pure Reason are according to the German Academy (Ak) edition pagination: Gesammelte Schriften, ed. Königlich Preußischen Akademie der Wissenschaften (Berlin, 1902–), later Deutschen Akademie der Wissenschaften zu Berlin. Passages from Kant’s Logic (edited by Kant’s student Jäsche and published under Kant’s name in 1800) are also cited by paragraph number (§) when appropriate. I use the translation in Lectures on Logic, ed. and trans. J. Michael Young (Cambridge: Cambridge University Press, 1992).

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De Morgan soon discovered, however, that the leading logician in Britain at the

time, William Hamilton, had himself been teaching that the traditional logic was

“perverted and erroneous in form.”4 In Germany, Maimon argued that Kant treated logic

as complete only because he omitted the most important part of critique – a critique of

logic itself.5 Hegel, less interested in formal logic than Maimon, concurs that “if logic has

not undergone any change since Aristotle, . . . then surely the conclusion which should be

drawn is that it is all the more in need of a total reconstruction.”6 On Hegel’s

reconstruction, logic “coincides with metaphysics.”7 Fries argued that Kant thought logic

complete only because he neglected “anthropological logic,” a branch of empirical

psychology that provides a theory of the capacities humans employ in thinking and a

basis for the meager formal content given in “demonstrative” logic.8 Trendelenburg later

argued that the logic contained in Kant’s Logic is not Aristotle’s logic at all, but a

corruption of it, since Aristotelian logic has metaphysical implications that Kant rejects.9

4 De Morgan, “Syllogism III,” 75. Cf. William Hamilton, Lectures on Logic, 2 vols., 3rd ed., ed. Mansel and Veitch (London, 1874), II, 251.

5 Solomon Maimon, Versuch einer neuen Logik oder Theorie des Denkens. Nebst angehängten Briefen des Philaletes an Aenesidemus (Berlin, 1794), 404ff. Partially translated by George di Giovanni as Essay towards a New Logic or Theory of Thought, Together with Letters of Philaletes to Aenesidemus, in Between Kant and Hegel, rev. ed. (Indianapolis: Hackett, 2000). Citations are from the pagination of the original German edition, which are reproduced in the English translation.

6 G. W. F. Hegel, Science of Logic, trans. A. V. Miller (London: George Allen & Unwin, 1969), 51. German edition: Gesammelte Werke (Hamburg: Felix Meiner, 1968–), 21:35. (Original edition, 1812–16, first volume revised in 1832.) I cite from the now-standard German edition, which contains the 1832 edition of “The Doctrine of Being,” the 1813 edition of “The Doctrine of Essence,” and the 1816 edition of “The Doctrine of the Concept” in vols. 21, 11, and 12, respectively.

7 G. W. F. Hegel, The Encyclopedia Logic: Part 1 of the Encyclopedia of Philosophical Sciences, trans. T. F. Geraets, W. A. Suchting, and H. S. Harris (Indianapolis: Hackett, 1991), §24. Gesammelte Werke (Hamburg: Felix Meiner, 1968–), 20: §24. (Original edition, 1817; second edition, 1830.) I cite from paragraph numbers (§) throughout.

8 Jakob Friedrich Fries, System der Logik, 3rd ed. (Heidelberg, 1837), 4–5. (Original edition, 1811.)

9 Adolf Trendelenburg, Logische Untersuchungen, 3rd ed., 2 vols. (Leipzig: S. Hirzel, 1870), I:32–3. (Original edition, 1840.)

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Indeed, one would be hard pressed to find a single nineteenth-century logician

who agrees with Kant’s notorious claim. However, this great expansion of logic – as

some logical works branched out into metaphysics, epistemology, philosophy of science,

and psychology, while others introduced new symbolic techniques and representations –

threatened to leave logicians with little common ground except for their rejection of

Kant’s conservatism. Robert Adamson, in his survey of logical history for the

Encyclopedia Britannica, writes of nineteenth-century logical works that “in tone, in

method, in aim, in fundamental principles, in extent of field, they diverge so widely as to

appear, not so many expositions of the same science, but so many different sciences.”10

Many historians of logic have understandably chosen to circumvent this problem by

ignoring many of the logical works that were the most widely read and discussed during

the period – the works of Hegel, Trendelenburg, Hamilton, Mill, Lotze, and Sigwart, for

example.

The present article, however, aims to be a history of “logic” in the multifaceted

ways in which this term was understood between Kant and Frege (though the history of

inductive logic – overlapping with the mathematical theory of probabilities and with

questions about scientific methodology – falls outside the purview of this article). There

are at least two reasons for this wide perspective. First, the diversity of approaches to

logic was accompanied by a continuous debate in the philosophy of logic over the nature,

extent, and proper method in logic. Second, the various logical traditions that coexisted in

the period – though at times isolated from one another – came to cross-pollinate with one

another in important ways. The first three sections of the article trace out the evolving

10 Robert Adamson, A Short History of Logic (London: W. Blackwood and Sons, 1911), 20. (Original edition, 1882.)

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conceptions of logic in Germany and Britain. The last three address the century’s most

significant debates over the nature of concepts, judgments and inferences, and logical

symbolism.

1. KANTIAN AND POST-KANTIAN LOGICS

Surprisingly, Kant was widely held in the nineteenth century to have been a logical

innovator. In 1912 Wilhelm Windelband wrote: “A century and a half ago, [logic] . . .

stood as a well-built edifice firmly based on the Aristotelian foundation. . . . But, as is

well-known, this state of things was entirely changed by Kant.”11 Kant’s significance

played itself out in two opposed directions: first, in his novel characterization of logic as

formal; and, second, in the new conceptions of logic advocated by those post-Kantian

philosophers who drew on Kant’s transcendental logic to attack Kant’s own narrower

conception of the scope of logic.

Though today the idea that logic is formal seems traditional or even definitional,

nineteenth-century logicians considered the idea to be a Kantian innovation.

Trendelenburg summarized the recent history:12

11 Wilhelm Windelband, Theories in Logic, trans. B. Ethel Meyer (New York: Citadel Press, 1961), 1. (Original edition, 1912.)

12 Contemporary Kant interpreters do not agree on whether Kant’s thesis that logic is formal was in fact novel. Some agree with Trendelenburg that Kant’s thesis was an innovative doctrine that depended crucially on other distinctive features of Kant’s philosophy; others think that Kant was reviving or modifying a traditional, Scholastic conception. (For an excellent recent paper that defends Trendelenburg’s view, see John MacFarlane, “Frege, Kant, and the Logic in Logicism,” Philosophical Review 111 [2000]: 25–65.) However, irrespective of the accuracy of this interpretive claim, the point remains that the thesis that logic is formal was considered by nineteenth-century logicians to be a Kantian innovation dependent on other parts of Kant’s philosophy. Besides Trendelenburg, see also Maimon, Logik, xx; William Hamilton, “Recent Publications on Logical Sciences,” reprinted in his Discussions on Philosophy and Literature, Education and University Reform (New York: Harper and Brothers, 1861), 145; J. S. Mill, An Examination of Sir William Hamilton’s Philosophy, reprinted as vol. 9 of Collected Works of John Stuart Mill, ed. J. M. Robson (Toronto: University of Toronto Press, 1979), 355; De Morgan, “Syllogism III,” 76; and Henry Mansel, Prolegomena Logica: An Inquiry into the

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Christian Wolff is still of the view that the grounds of logic derive from ontology

and psychology and that logic precedes them only in the order in which the

sciences are studied. For the first time in Kant’s critical philosophy, in which the

distinction of matter and form is robustly conceived, formal logic clearly emerges

and actually stands and falls with Kant.13

General logic for Kant contains the “absolutely necessary rules of thinking,

without which no use of the understanding takes place” (A52/B76). The understanding –

which Kant distinguishes from “sensibility” – is the faculty of “thinking,” or “cognition

through concepts” (A50/B74; Ak 9:91). Unlike Wolff, Kant claims a pure logic “has no

empirical principles, thus it draws nothing from psychology” (A54/B78). The principles

of psychology tell how we do think; the principles of pure general logic, how we ought to

think (Ak 9:14). The principles of logic do not of themselves imply metaphysical

principles; Kant rejects Wolff and Baumgarten’s proof of the principle of sufficient

reason from the principle of contradiction (Ak 4:270).14 Though logic is a canon, a set of

rules, it is not an organon, a method for expanding our knowledge (Ak 9:13).

For Kant, pure general logic neither presupposes nor of itself implies principles of

any other science because it is formal. “General logic abstracts . . . from all content of

cognition, i.e., from any relation of it to the object, and considers only the logical form in

the relation of cognitions to one another” (A55/B79). In its treatment of concepts, formal

logic takes no heed of the particular marks that a given concept contains, nor of the

Psychological Character of Logical Processes, 2nd ed. (Boston, 1860), iv–v. Among these logicians, Hamilton was unique in recognizing the affinity between Kant’s conception of logic and Scholastic notions.

13 Trendelenburg, Untersuchungen, I, 15. In this passage, Trendelenburg cites Christian Wolff, Philosophia rationalis sive Logica (Frankfurt, 1728), §88–9.

14 See Christian Wolff, Philosophia prima sive ontologia (Frankfurt, 1730), §70. Alexander Gottlieb Baumgarten, Metaphysica, 3rd ed. (Halle, 1757).

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particular objects that are contained under it. In its treatment of judgments, formal logic

attends merely to the different ways in which one concept can be contained in or under

one another. (So in a judgment like “All whales are mammals,” the word “all” and the

copula “is” do not represent concepts, but express the form of the judgment, the particular

way in which a thinker combines the concepts whale and mammal.)

The generality of logic requires this kind of formality because Kant, as an

essential part of his critique of dogmatic metaphysicians such as Leibniz and Wolff,

distinguishes mere thinking from cognizing (or knowing) (B146). Kant argues against

traditional metaphysics that, since we can have no intuition of noumena, we cannot have

cognitions or knowledge of them. But we can coherently think noumena (B166n). This

kind of thinking is necessary for moral faith, where the subject is not an object of

intuition, but, for example, the divine being as moral lawgiver and just judge. Thus,

formal logic, which abstracts from all content of cognition, makes it possible for us

coherently to conceive of God and things in themselves.

The thesis of the formality of logic, then, is intertwined with some of the most

controversial aspects of the critical philosophy: the distinction between sensibility and

understanding, appearances and things in themselves. Once these Kantian “dualisms”

came under severe criticism, post-Kantian philosophers also began to reject the

possibility of an independent formal logic.15 Hegel, for example, begins his Science of

15 A classic attack on Kant’s distinction between sensibility and understanding is Solomon Maimon, Versuch über die Transcendentalphilosophie, reprinted in Gesammelte Werke, vol. 2, ed. Valerio Verra (Hildsheim: Olms, 1965–76), 63–4. A classic attack on Kant’s distinction between appearances and things in themselves is F. H. Jacobi, David Hume über den Glauben, oder Idealismus und Realismus. Ein Gespräch (Breslau, 1787). I have emphasized that Kant defends the coherence of the doctrine of unknowable things in themselves by distinguishing between thinking and knowing – where thinking, unlike knowing, does not require the joint operation of sensibility and understanding. Formal logic, by providing rules for the use of the understanding and abstracting from all content provided by sensibility, makes room for the

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Logic with a polemic against Kant’s conception of formal logic: if there are no

unknowable things in themselves, then the rules of thinking are rules for thinking an

object, and the principles of logic become the first principles of ontology.16

Further, Kant’s insistence that the principles of logic are not drawn from

psychology or metaphysics leaves open a series of epistemological questions. How then

do we know the principle of contradiction? How do we know that there are precisely

twelve logical forms of judgment? Or that some figures of the syllogism are valid and

others not? Many agreed with Hegel that Kant’s answers had “no other justification than

that we find such species already to hand and they present themselves empirically.”17

Kant’s unreflective procedure endangers both the a priori purity and the certainty of

logic.18

Though Kant says only that “the labors of the logicians were ready to hand” (Ak

4:323), his successors were quick to propose novel answers to these questions. Fries

appealed to introspective psychology, and he reproved Kant for overstating the

independence of “demonstrative logic” from anthropology.19 Others grounded formal

idea that we can coherently think of things in themselves. Now, the fact that Kant defends the coherence of his more controversial doctrines by appealing to the formality of logic does not yet imply that an attack on Kantian “dualisms” need also undermine the thesis that logic is formal. But, as we will see, many post-Kantian philosophers thought that an attack on Kant’s distinctions would also undermine the formality thesis – or at least they thought that such an attack would leave the formality of logic unmotivated.

16 Hegel, Science of Logic, 43–8 (Werke 21:28–32).17 Hegel, Science of Logic, 613 (Werke 12:43).18 See also J. G. Fichte, “First Introduction to the Science of Knowledge,” trans. Peter Heath and

John Lachs in The Science of Knowledge (New York: Cambridge University Press, 1982), I 442. See Fichtes Werke, ed. I. H. Fichte, Band I (Berlin: Walter de Gruyter, 1971). (Original edition of “First Introduction,” 1797.) Citations are by volume and page number from I. H. Fichte’s edition, which are reproduced in the margins of the English translations. K. L. Reinhold, The Foundation of Philosophical Knowledge, trans. George di Giovanni in Between Kant and Hegel, rev. ed. (Indianapolis: Hackett, 2000), 119. Citations follow the pagination of the original 1794 edition, Über das Fundament des philosophischen Wissens (Jena, 1794), which are reproduced in the margins of the English edition.

19 Fries, Logik, 5.

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logic in what Kant called “transcendental logic.” Transcendental logic contains the rules

of a priori thinking (A57/B81). Since all use of the understanding, inasmuch as it is

cognizing an object, requires a priori concepts (the categories), transcendental logic then

expounds also “the principles without which no object can be thought at all” (A62/B87).

Reinhold argued – against Kant’s “Metaphysical Deduction” of the categories from the

forms of judgments – that the principles of pure general logic should be derived from a

transcendental principle (such as his own principle of consciousness).20 Moreover, logic

can only be a science if it is systematic, and this systematicity (on Reinhold’s view)

requires that logic be derived from an indemonstrable first principle.21

Maimon’s 1794 Versuch einer neuen Logik oder Theorie des Denkens, which

contains both an extended discussion of formal logic and an extended transcendental

logic, partially carries out Reinhold’s program. There are two highest principles: the

principle of contradiction (which is the highest principle of all analytic judgments) and

Maimon’s own “principle of determinability” (which is the highest principle of all

synthetic judgments) (19–20). Since formal logic presupposes transcendental logic (xx–

xxii), Maimon defines the various forms of judgment (such as affirmative and negative)

using transcendental concepts (such as reality and negation). He proves various features

of syllogisms (such as that the conclusion of a valid syllogism is affirmative iff both its

premises are) using the transcendental principle of determinability (94–5).

Hegel’s Science of Logic is surely the most ambitious and influential of the

logical works that include both formal and transcendental material.22 However, Hegel 20 Reinhold, Foundation, 118–21.21 See also Fichte, “Concerning the Concept of the Wissenschaftslehre,” trans. Daniel Breazeale

in Fichte: Early Philosophical Writings (Ithaca, N.Y.: Cornell University Press, 1988), I 41–2. (Original edition, 1794.)

22 Hegel, Science of Logic, 62 (Werke 21:46).

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cites as a chief inspiration – not Maimon, but – Fichte. For Hegel, Fichte’s philosophy

“reminded us that the thought-determinations must be exhibited in their necessity, and

that it is essential for them to be deduced.”23 In Fichte’s Wissenschaftslehre, the whole

system of necessary representations is deduced from a single fundamental and

indemonstrable principle.24 In his 1794 book, Foundations of the Entire Science of

Knowledge, Fichte derives from the first principle “I am I” not only the category of

reality, but also the logical law “A = A”; in subsequent stages he derives the category of

negation, the logical law “~A is not equal to A, ” and finally even the various logical

forms of judgment. Kant’s reaction, given in his 1799 “Open Letter on Fichte’s

Wissenschaftslehre,” is unsurprising: Fichte has confused the proper domain of logic with

metaphysics (Ak 12:370–1). Later, Fichte follows the project of the Wissenschaftslehre to

its logical conclusion: transcendental logic “destroys” the common logic in its

foundations, and it is necessary to refute (in Kant’s name) the very possibility of formal

logic.25

Hegel turned Kant’s criticism of Fichte on its head: were the critical philosophy

consistently thought out, logic and metaphysics would in fact coincide. For Kant, the

understanding can combine or synthesize a sensible manifold, but it cannot itself produce

the manifold. For Hegel, however, there can be an absolute synthesis, in which thinking

itself provides contentful concepts independently of sensibility.26 Kant had argued that if

the limitations of thinking are disregarded, reason falls into illusion. In a surprising twist,

23 Hegel, Encyclopedia Logic, §42.24 Fichte, “First Introduction,” I 445–6.25 Fichte, “Ueber das Verhältniß der Logik zur Philosophie oder transscendentale Logik,”

reprinted in Fichtes Werke, ed. I. H. Fichte, Band IX (Berlin: Walter de Gruyter, 1971), IX 111–12. This text is from a lecture course delivered in 1812.

26 Hegel, Faith and Knowledge, trans. W. Cerf and H. Harris (Albany, N.Y.: SUNY Press, 1977), 72 (Werke 4:328). (Original edition, 1802.)

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Hegel uses this dialectical nature of pure reason to make possible his own non-Kantian

doctrine of synthesis. Generalizing Kant’s antinomies to all concepts, Hegel argues that

any pure concept, when thought through, leads to its opposite.27 This back-and-forth

transition from a concept to its opposite – which Hegel calls the “dialectical moment” –

can itself be synthesized into a new unitary concept – which Hegel calls the “speculative

moment,” and the process can be repeated.28 Similarly, the most immediate judgment, the

positive judgment (e.g., “The rose is red”), asserts that the individual is a universal. But

since the rose is more than red, and the universal red applies to more than the rose, the

individual is not the universal, and we arrive at the negative judgment.29 Hegel iterates

this procedure until he arrives at a complete system of categories, forms of judgment,

logical laws, and forms of the syllogism. Moreover, by beginning with the absolutely

indeterminate and abstract thought of being,30 he has rejected Reinhold’s demand that

logic begin with a first principle: Hegel’s Logic is “preceded by . . . total

presuppositionlessness.”31

2. THE REVIVAL OF LOGIC IN BRITAIN

The turnaround in the fortunes of logic in Britain was by near consensus attributed to

Whately’s 1826 Elements of Logic.32 A work that well illustrates the prevailing view of

logic in the Anglophone world before Whately is Harvard Professor of Logic Levi

27 Hegel, Encyclopedia Logic, §48.28 Hegel, Encyclopedia Logic, §§81–2.29 Hegel, Science of Logic, 594, 632ff. (Werke 12:27, 61ff.).30 Hegel, Science of Logic, 70 (Werke 21:58–9).31 Hegel, Encyclopedia Logic, §78.32 See De Morgan, “Logic,” reprinted in On the Syllogism, 247. (De Morgan’s essay first

appeared in 1860.) Hamilton, “Recent Publications,” 128. (Original edition, 1833.) John Stuart Mill, System of Logic, Ratiocinative and Inductive, reprinted as vols. 7 and 8 of Collected Works of John Stuart Mill, ed. J. M. Robson (Toronto: University of Toronto Press, 1973), 7: cxiv. (Original edition, 1843.)

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Hedge’s Elements of Logick. The purpose of logic, Hedge claims, is to direct the

intellectual powers in the investigation and communication of truths.33 This means that a

logical treatise must trace the progress of knowledge from simple perceptions to the

highest discoveries of reasoning. The work thus reads more like Locke’s Essay than

Kant’s Logic – it draws heavily not only on Locke, but also on Reid and on Hume’s laws

of the associations of ideas. Syllogistic, however, is discussed only in a footnote – since

syllogistic is “of no use in helping us to the discovery of new truths.”34 Hedge here cites

Locke’s Essay (IV.xvii.4), where Locke argued that syllogistic is not necessary for

reasoning well – “God has not been so sparing to Men to make them barely two-legged

Creatures, and left it to Aristotle to make them Rational.” Syllogisms, Locke claims, are

of no use in the discovery of new truths or the finding of new proofs; indeed, they are

inferior in this respect to simply arranging ideas “in a simple and plain order.”

For Whately, Locke’s objection that logic is unserviceable in the discovery of the

truth misses the mark, because it assumes a mistaken view of logic.35 The chief error of

the “schoolmen” was the unrealizable expectation they raised: that logic would be an art

that furnishes the sole instrument for the discovery of truth, that the syllogism would be

an engine for the investigation of nature (viii, 4, 5). Fundamentally, logic is a science and

not an art (1). Putting an argument into syllogistic form need not add to the certainty of

the inference, any more than natural laws make it more certain that heavy objects fall.

Indeed, Aristotle’s dictum de omni et nullo is like a natural law: it provides an account of

the correctness of an argument; it shows us the one general principle according to which

takes place every individual case of correct reasoning. A logician’s goal then is to show 33 Levi Hedge, Elements of Logick (Cambridge, 1816), 13.34 Hedge, Logick, 152, 148.35 Richard Whately, Elements of Logic, 9th ed. (London, 1866), 15. (Original edition, 1826.)

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that all correct reasoning is conducted according to one general principle – Aristotle’s

dictum – and is an instance of the same mental process – syllogistic (75).

In Hamilton’s wide-ranging and erudite review of Whately’s Elements, he

acknowledged that Whately’s chief service was to correct mistakes about the nature of

logic, but he excoriated his fellow Anglophones for their ignorance of historical texts and

contemporary German logics. Indeed, we can more adequately purify logic of intrusions

from psychology and metaphysics and more convincingly disabuse ourselves of the

conviction that logic is an “instrument of scientific discovery” by accepting Kant’s idea

that logic is formal.36 Hamilton’s lectures on logic, delivered in 1837–8 using the German

Kantian logics written by Krug and Esser,37 thus introduced into Britain the Kantian idea

that logic is formal.38 For him, the form of thought is the kind and manner of thinking an

object (I 13) or the relation of the subject to the object (I 73). He distinguishes logic from

psychology (against Whately) as the science of the product, not the process, of thinking.

Since the forms of thinking studied by logic are necessary, there must be laws of thought:

the principles of identity, contradiction, and excluded middle (I 17, II 246). He

distinguishes physical laws from “formal laws of thought,” which thinkers ought to –

though they do not always – follow (I 78).39

Mill later severely (and justly) criticized Hamilton for failing to characterize the

distinction between the matter and form of thinking adequately. Mill argued that it is

impossible to take over Kant’s matter/form distinction without also taking on all of 36 Hamilton, “Recent Publications,” 139.37 Wilhelm Traugott Krug, Denklehre oder Logik (Königsberg, 1806). Jakob Esser, System der

Logik, 2nd ed. (Münster, 1830). (Original edition, 1823.)38 Hamilton, Logic. I cite from the 1874 3rd ed. (Original edition, 1860.)39 Hamilton unfortunately muddies the distinction between the two kinds of laws by lining it up

with the Kantian distinction between the doctrine of elements and the doctrine of method (Logic, I 64).

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Kant’s transcendental idealism.40 Mansel tries to clarify the distinction between matter

and form by arguing that the form of thinking is expressed in analytic judgments. He

claims (as neither Hamilton nor Kant himself had done explicitly) that the three laws of

thought are themselves analytic judgments and that the entire content of logic is derivable

from these three laws.41 Moreover, Mansel further departs from Kant and Hamilton by

restricting the task of logic to characterizing the form and laws of only analytic

judgments.42

In his 1828 review, Mill criticized Whately for concluding that inductive logic –

that is, the rules for the investigation and discovery of truth – could never be put into a

form as systematic and scientific as syllogistic.43 Mill’s System of Logic, Ratiocinative

and Inductive, which was centered around Mill’s famous five canons of experimental

inquiry, aimed to do precisely what Whately thought impossible. The work, which

included material we would now describe as philosophy of science, went through eight

editions and became widely used in colleges throughout nineteenth-century Britain.

Logic for Mill is the science as well as the art of reasoning (5); it concerns the operations

of the understanding in giving proofs and estimating evidence (12). Mill argued that in

fact all reasoning is inductive (202). There is an inconsistency, Mill alleges, in thinking

that the conclusion of a syllogism (e.g., “Socrates is mortal”) is known on the basis of the

premises (e.g., “All humans are mortal” and “Socrates is human”), while also admitting

40 Mill, Examination, 355. (Original edition, 1865.)41 Mansel, Prolegomena, 159. (Original edition, 1851.) A similar position had been defended

earlier by Fries, Logik, §40. Mansel almost certainly got the idea from Fries.42 Mansel, Prolegomena, 202. Although Kant did say that the truth of an analytic judgment could

be derived entirely from logical laws (see, e.g., A151/B190–1), he always thought that logic considered the forms of all judgments whatsoever, whether analytic or synthetic.

43 John Stuart Mill, “Whately’s Elements of Logic,” reprinted in vol. 11 of Collected Works of John Stuart Mill, ed. J. M. Robson (Toronto: University of Toronto Press, 1978). Cf. Whately, Elements, 168; Mill, Logic, cxii.

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that the syllogism is vicious if the conclusion is not already asserted in the premises

(185). Mill’s solution to this paradox is that the syllogistic inference is only apparent: the

real inference is the induction from the particular facts about the mortality of particular

individuals to the mortality of Socrates. The inference is then actually finished when we

assert, “All men are mortal” (186–7).

The debate over whether logic is an art and the study of logic useful for reasoning

dovetailed with concurrent debates over curricular reform at Oxford. By 1830, Oxford

was the only British institution of higher learning where the study of logic had survived.44

Some, such as Whewell, advocated making its study elective, allowing students to train

their reasoning by taking a course on Euclid’s Elements.45 Hamilton opposed this

proposal,46 as did the young mathematician Augustus De Morgan, who thought that the

study of syllogistic facilitates a student’s understanding of geometrical proofs.47 Indeed,

De Morgan’s first foray into logical research occurred in a mathematical textbook, where,

in a chapter instructing his students on putting Euclidean proofs into syllogistic form, he

noticed that some proofs require treating “is equal to” as a copula distinct from “is,”

though obeying all of the same rules.48

These reflections on mathematical pedagogy led eventually to De Morgan’s

logical innovations. In his Formal Logic,49 De Morgan noted that the rules of the

44 Hamilton, “Recent Publications,” 124. De Morgan, “On the Methods of Teaching the Elements of Geometry,” Quarterly Journal of Education 6 (1833): 251.

45 W. Whewell, Thoughts on the Study of Mathematics as Part of a Liberal Education (Cambridge, 1835).

46 Hamilton, “On the Study of Mathematics, as an Exercise of Mind,” reprinted in Discussions on Philosophy and Literature, Education and University Reform (New York, 1861). (Original edition, 1836.)

47 De Morgan, “Methods of Teaching,” 238–9.48 De Morgan, On the Study and Difficulties of Mathematics (London, 1831).49 De Morgan, Formal Logic (London, 1847).

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syllogism work for copulae other than “is” – as long as they have the formal properties of

transitivity, reflexivity, and what De Morgan calls “contrariety” (57–9). Transitivity is the

common form, and what distinguishes “is” from “is equal to” is their matter. This

generalization of the copula culminated in De Morgan’s paper “On the Syllogism IV,”

the first systematic study of the logic of relations.50 He considers propositions of the form

“A..LB” (“A is one of the Ls of B”), where “L” denotes any relation of subject to

predicate. He thinks of “A” as the subject term, “B” as the predicate term, and “..L” as the

relational expression that functions as the copula connecting subject and predicate.51

Thus, if we let “L” mean “loves,” then “A..LB” would mean that “A is one of the lovers

of B” – or just “A loves B.” He symbolized contraries using lowercase letters: “A..lB”

means “A is one of the nonlovers of B” or “A does not love B.” Inverse relations are

symbolized using the familiar algebraic expression: “A..L-1B” means “A is loved by B.”

Importantly, De Morgan also considered compound relations, or what we would now call

“relative products”: “A..LPB” means “A is a lover of a parent of B.” De Morgan

recognized, moreover, that reasoning with compound relations required some simple

quantificational distinctions. We symbolize “A is a lover of every parent of B” by adding

an accent: “A.. LP'B.” De Morgan was thus able to state some basic facts and prove some

theorems about compound relations. For instance, the contrary of LP is lP' and the

converse of the contrary of LP is p-1L-1’.52

50 De Morgan, “On the Syllogism IV,” reprinted in On the Syllogism. (Original edition, 1860.)51 The two dots preceding “L” indicate that the statement is affirmative; an odd number of dots

indicates that the proposition is negative.52 Spelled out a bit (and leaving off the quotation marks for readability): that LP is the contrary

of lP' means that A..LPB is false iff A..lP'B is true. That is, A does not love any of B’s parents iff A is a nonlover of every parent of B. That the converse of the contrary of LP is p-1L-1' means that A..LPB is false iff B..p-1L-1'A is true. That is, A does not love any of B’s parents iff B is not the child of anyone A loves.

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De Morgan recognized that calling features of the copula “is” material departed

from the Kantian view that the copula is part of the form of a judgment.53 De Morgan,

however, thought that the logician’s matter/form distinction could be clarified by the

mathematician’s notion of form.54 From the mathematician’s practice we learn two

things. First, the form/matter distinction is relative to one’s level of abstraction: the

algebraist’s x + y is formal with respect to 4 + 3, but x + y as an operation on numbers is

distinguished only materially from the similar operations done on vectors or differential

operators.55 Second, the form of thinking is best understood on analogy with the principle

of a machine in operation.56

In thinking of mathematics as a mechanism, De Morgan is characterizing

mathematics as fundamentally a matter of applying operations to symbols according to

laws of their combinations. Here De Morgan is drawing on work done by his fellow

British algebraists. (In fact, De Morgan’s logical work is the confluence of three

independent intellectual currents: the debate raging from Locke to Whately over the value

of syllogistic, the German debate – imported by Hamilton – over Kant’s matter/form

distinction,57 and the mathematical debate centered at Cambridge over the justification of

certain algebraic techniques.) As a rival to Newton’s geometric fluxional calculus, the

Cambridge “Analytical Society” together translated Lacroix’s An Elementary Treatise on 53 De Morgan, “On the Syllogism II,” reprinted in On the Syllogism, 57–8. (Original edition,

1850.)54 De Morgan, “Syllogism III,” 78.55 De Morgan, “Logic,” 248; “Syllogism III,” 78.56 De Morgan, “Syllogism III,” 75.57 We saw earlier that Mill argued that one could not maintain that logic is “formal” unless one

were willing to take on Kant’s matter/form distinction – and therefore also Kant’s transcendental idealism. De Morgan, on the other hand, thought that introducing the mathematician’s notion of “form” would allow logicians to capture what is correct in Kant’s idea that logic is formal, but without having to take on the rest of the baggage of Kantianism. However, the effect of making use of this notion of form taken from British algebra is – for better or worse – to transform Kant’s way of conceiving the formality of logic.

16

the Differential and Integral Calculus, a calculus text that contained, among other

material, an algebraic treatment of the calculus drawing on Lagrange’s 1797 Théorie des

fonctions analytiques. Lagrange thought that every function could be expanded into a

power series expansion, and its derivative defined purely algebraically. Leibniz’s “dx/dy”

was not thought of as a quotient, but as “a differential operator” applied to a function.

These operators could then be profitably thought of as mathematical objects subject to

algebraic manipulation58 – even though differential operators are neither numbers nor

geometrical magnitudes. This led algebraists to ask just how widely algebraic operations

could be applied, and to ask after the reason for their wide applicability. (And these

questions would be given a very satisfactory answer if logic itself were a kind of algebra.)

A related conceptual expansion of algebra resulted from the use of negative and

imaginary numbers. Peacock’s A Treatise on Algebra provided a novel justification: he

distinguished “arithmetical algebra” from “symbolic algebra” – a strictly formal science

of symbols and their combinations, where “a – b” is meaningful even if a < b. Facts in

arithmetical algebra can be transferred into symbolic algebra by the “principle of the

permanence of equivalent forms.”59 Duncan Gregory defined symbolic algebra as the

“science which treats of the combination of operations defined not by their nature, that is,

by what they are or what they do, but by the laws of combination to which they are

58 As did Servois – see F. J. Servois, Essai sur un nouveau mode d'exposition des principes du calcul différentiel (Paris, 1814).

59 George Peacock, A Treatise on Algebra, vol. I, Arithmetical Algebra; vol. II, On Symbolical Algebra and Its Applications to the Geometry of Position, 2nd ed. (Cambridge, 1842–5), II 59. (Original edition, 1830.)

17

subject.”60 This is the background to De Morgan’s equating the mathematician’s notion of

form with the operation of a mechanism.

Gregory identifies five different kinds of symbolic algebras.61 One algebra is

commutative, distributive, and subject to the law am⋅an = am+n. George Boole, renaming

the third law the “index law,” followed Gregory in making these three the fundamental

laws of the algebra of differential operators.62 Three years later Boole introduced an

algebra of logic that obeys these same laws, with the index law modified to xn = x for n

nonzero.63 The symbolic algebra defined by these three laws could be interpreted in

different ways: as an algebra of the numbers 0 and 1, as an algebra of classes, or as an

algebra of propositions.64 Understood as classes, ab is the class of things that are in a and

in b; a + b is the class of things that are in a or in b, but not in both; and a – b is the class

of things that are in a and not in b; 0 and 1 are the empty class and the universe. The

index law holds (e.g., for n = 2) because the class of things that are in both x and x is just

x. These three laws are more fundamental than Aristotle’s dictum;65 in fact, the principle

of contradiction is derivable from the index law, since x – x2 = x(1 – x) = 0.66

The class of propositions in Boole’s algebra – equations with arbitrary numbers of

class terms combined by multiplication, addition, and subtraction – is wider than the class

amenable to syllogistic, which only handles one subject and one predicate class per

60 Duncan Gregory, “On the Real Nature of Symbolical Algebra,” reprinted in The Mathematical Works of Duncan Farquharson Gregory, ed. William Walton (Cambridge, 1865), 2. (Original edition, 1838.)

61 Gregory, “Symbolical Algebra,” 6–7.62 George Boole, “On a General Method in Analysis,” Philosophical Transactions of the Royal

Society of London 134 (1844): 225–82.63 Boole, The Mathematical Analysis of Logic (Cambridge, 1847), 16–18.64 Boole, An Investigation of the Laws of Thought (London, 1854), 37.65 Boole, Mathematical Analysis, 18.66 Boole, Laws of Thought, 49.

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proposition.67 Just as important, Boole can avoid all of the traditional techniques of

conversion, mood, and figure by employing algebraic techniques for the solution of

logical equations. Despite the impressive power of Boole’s method, it falls well short of

modern standards of rigor. The solution of equations generally involves eliminating

factors, and so dividing class terms – even though Boole admits that no logical

interpretation can be given to division.68 But since Boole allows himself to treat logical

equations as propositions about the numbers 0 and 1, the interpretation of division is

reduced to interpreting the coefficients “1/1,” “1/0,” “0/1,” and “0/0.” Using informal

justifications that convinced few, he rejected “1/0” as meaningless and threw out every

term with it as a coefficient, and he interpreted “0/0” as referring to some indefinite class

“v.”

Boole, clearly influenced by Peacock, argued that there was no necessity in giving

an interpretation to logical division, since the validity of any “symbolic process of

reasoning” depends only on the interpretability of the final conclusion.69 Jevons thought

this an incredible position for a logician and discarded division in order to make all

results in his system interpretable.70 Venn retained logical division but interpreted it as

“logical abstraction” –as had Schröder, whose 1877 book introduced Boolean logic into

Germany.71 Boole had interpreted disjunction exclusively, allowing him to interpret his

equations indifferently as about classes or the algebra of the numbers 0 and 1 (for which

67 Boole, Laws of Thought, 238.68 Boole, Laws of Thought, 66ff.69 Boole, Laws of Thought, 66ff.70 William Stanley Jevons, “Pure Logic or The Logic of Quality apart from Quantity, with

Remarks on Boole’s System and on the Relation of Logic to Mathematics,” reprinted in Pure Logic and Other Minor Works, ed. Robert Adamson and Harriet A. Jevons (New York, 1890), §174 and §197ff. (Original edition, 1864.)

71 Venn, Symbolic Logic, 73ff. Ernst Schröder, Der Operationskreis des Logikkalküls (Leipzig, 1877), 33.

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“x + y” has meaning only if xy = 0). Jevons argued that “or” in ordinary language is

actually inclusive, and he effected great simplification in his system by introducing the

law A + A = A.72

Peirce departs from Boole’s inconvenient practice of only considering equations

and introduces a primitive symbol for class inclusion. Peirce also combines in a fruitful

way Boole’s algebra with De Morgan’s logic of relations. He conceives of relations (not

as copulae, but) as classes of ordered pairs,73 and he introduces Boolean operations on

relations. Thus “l + s” means “lover or servant.”74 This research culminated in Peirce’s

1883 “The Logic of Relatives,” which independently of Frege introduced into the

Boolean tradition polyadic quantification.75 Peirce writes the relative term “lover” as

l = ∑i∑j (l)ij(I : J)

where “∑” denotes the summation operator, “I:J” denotes an ordered pair of individuals,

and “(l)ij” denotes a coefficient whose value is 1 if I does love J, and 0 otherwise (195).

Then, for instance,

∏i∑j (l)ij > 0

means “everybody loves somebody” (207).

72 Jevons, “Pure Logic,” §178, §193; cf. Schröder, Operationskreis, 3. Though Jevons’s practice of treating disjunction inclusively has since become standard, Boole’s practice was in keeping with the tradition. Traditional logicians tended to think of disjunction on the model of the relation between different subspecies within a genus, and since two subspecies exclude one another, so too did disjunctive judgments. (See, e.g., Kant A73–4/B99.)

73 Charles Sanders Peirce, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic,” reprinted in Collected Papers of Charles Sanders Peirce. Vol. 3, Exact Logic, ed. Charles Hartshorne and Paul Weiss (Cambridge: Harvard University Press, 1933), 76. (Original edition, 1870.)

74 Peirce, “Description,” 38.75 Peirce, “The Logic of Relatives,” reprinted in Collected Papers, vol. 3. (Original edition,

1883.)

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3. GERMAN LOGIC AFTER HEGEL

As post-Kantian idealism waned after Hegel’s death, the most significant German

logicians – Trendelenburg, Lotze, Sigwart, and Ueberweg – came to prefer a middle way

between a “subjectively formal” logic and an identification of logic with metaphysics.76

On this view, logic is not the study of mere thinking or of being itself, but of knowledge –

a position articulated earlier in Friedrich Schleiermacher’s Dialektik.

In lectures given between 1811 and 1833, Schleiermacher calls his “dialectic” the

“science of the supreme principles of knowing” and “the art of scientific thinking.”77 To

produce knowledge is an activity, and so the discipline that studies that activity is an art,

not a mere canon.78 This activity is fundamentally social and occurs within a definite

historical context. Schleiermacher thus rejects the Fichtean project of founding all

knowledge on a first principle; instead, our knowledge always begins “in the middle.”79

Because individuals acquire knowledge together with other people, dialectic is also –

playing up the Socratic meaning – “the art of conversation in pure thinking.”80 Though

transcendental and formal philosophy are one,81 the principles of being and the principles

of knowing are not identical. Rather, there is a kind of parallelism between the two

76 See Friedrich Ueberweg, System of Logic and History of Logical Doctrines, trans. Thomas M. Lindsay (London, 1871), §1, §34. System der Logik und Geschichte der logischen Lehren, 3rd ed. (Bonn, 1868). (Original edition, 1857.)

77 Friedrich Schleiermacher, Dialectic or The Art of Doing Philosophy, ed. and trans. T. N. Tice (Atlanta: Scholar's Press, 1996), 1–5. Dialektik (1811), ed. Andreas Arndt (Hamburg: Felix Meiner, 1986), 3–5, 84. These books contain lecture notes from Schleiermacher’s first lecture course.

78 Schleiermacher, Dialektik, ed. L. Jonas as vol. 4 of the second part of Friedrich Schleiermacher’s sämmtliche Werke. Dritte Abtheilung: Zur Philosophie (Berlin, 1839), 364. The Jonas edition of Dialektik contains notes and drafts produced between 1811 and 1833. Schleiermacher’s target here is, of course, Kant, who thought that though logic is a set of rules (a “canon”), it is not an “organon” – an instrument for expanding our knowledge.

79 Schleiermacher, Dialectic, 3 (Dialektik (1811), 3). Schleiermacher, Dialektik, §291.80 Schleiermacher, Dialektik, §1.81 Schleiermacher, Dialectic, 1–2 (Dialektik (1811), 5).

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realms. For example, corresponding to the fact that our thinking employs concepts and

judgments is the fact that the world is composed of substantial forms standing in

systematic causal relations.82

Trendelenburg, whose enthusiasm for Aristotle’s logic over its modern

perversions led him to publish a new edition of Aristotle’s organon for student use,83

agreed with Schleiermacher that logical principles correspond to, but are not identical

with, metaphysical principles. But unlike Schleiermacher,84 Trendelenburg thought that

syllogisms are indispensable for laying out the real relations of dependence among things

in nature, and he argued that there is a parallelism between the movement from premise

to conclusion and the movements of bodies in nature.85

In 1873 and 1874 Christoph Sigwart and Hermann Lotze produced the two

German logic texts that were perhaps most widely read in the last decades of the

nineteenth century – a period that saw a real spike in the publication of new logic texts.

Sigwart sought to “reconstruct logic from the point of view of methodology.”86 Lotze’s

Logic begins with an account of how the operations of thought allow a subject to

apprehend truths.87 In the current of ideas in the mind, some ideas flow together only

because of accidental features of the world; some ideas flow together because the realities

82 Schleiermacher, Dialektik, §195.83 Adolf Trendelenburg, Excerpta ex Organo Aristotelis (Berlin, 1836).84 For Schleiermacher, syllogistic is not worth studying, since no new knowledge can arise

through syllogisms. See Schleiermacher, Dialectic, 36 (Dialektik (1811), 30); Schleiermacher, Dialektik, §§327–9.

85 Trendelenburg, Untersuchungen, II 388; I 368.86 Christoph Sigwart, Logic, 2 vols., trans. Helen Dendy (New York, 1895), I i. Logik. 2 vols.,

2nd ed. (Freiburg, 1889, 1893). (1st ed. of vol. 1, 1873; 2nd ed., 1878.)87 Hermann Lotze, Logic, 2 vols., trans. Bernard Bosanquet (Oxford, 1884), §II. Logik. Drei

Bücher vom Denken, vom Untersuchen und vom Erkennen, 2nd ed. (Leipzig, 1880). (Original edition, 1874.) Whenever possible, I cite by paragraph numbers (§), which are common to the English and German editions.

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that give rise to them are in fact related in a nonaccidental way. It is the task of thought to

distinguish these two cases – to “reduce coincidence to coherence” – and it is the task of

logic to investigate how the concepts, judgments, and inferences of thought introduce this

coherence.88

The debate over the relation between logic and psychology, which had been

ongoing since Kant, reached a fever pitch in the Psychologismus-Streit of the closing

decades of the nineteenth century. The term “psychologism” was coined by the Hegelian

J. E. Erdmann to describe the philosophy of Friedrich Beneke.89 Erdmann had earlier

argued in his own logical work that Hegel, for whom logic is presuppositionless, had

decisively shown that logic in no way depends on psychology – logic is not, as Beneke

argued, “applied psychology.”90

Contemporary philosophers often associate psychologism with the confusion

between laws describing how we do think and laws prescribing how we ought to think.91

This distinction appears in Kant (Ak 9:14) and was repeated many times throughout the

century.92 The psychologism debate, however, was not so much about whether such a

distinction could be drawn, but whether this distinction entailed that psychology and

logic were independent. Husserl argued that the distinction between descriptive and

normative laws was insufficient to protect against psychologism, since the psychologistic

88 Lotze, Logic, §XI.89 Johann Eduard Erdmann, Grundriss der Geschichte der Philosophie. Zweiter Band, 2nd ed.

(Berlin, 1870), 636.90 Johann Eduard Erdmann, Outlines of Logic and Metaphysics, trans. B. C. Burt (New York,

1896), §2. Grundriss der Logik und Metaphysik, 4th ed. (Halle, 1864). (Original edition, 1841.) For a representative passage in Beneke, see System der Logik als Kunstlehre des Denken. Erster Theil (Berlin, 1842), 17–18.

91 See, for example, Gottlob Frege, Grundgesetze der Arithmetik, vol. 1 (Jena: H. Pohle, 1893), xv, trans. Michael Beaney in The Frege Reader (Oxford: Blackwell Publishers, 1997), 202.

92 See, for example, Johann Friedrich Herbart, Psychologie als Wissenschaft (Königsberg, 1825), 173; Boole, Laws of Thought, 408–9.

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logician could maintain, as did Mill, that the science of correct reasoning is a branch of

psychology, since the laws that describe all thinking surely also apply to the subclass of

correct thinking.93 Indeed, Mill argues, if logic is to characterize the process of correct

thinking, it has to draw on an analysis of how human thinking in fact operates.94

Mansel argued that the possibility of logical error in no way affects the character

of logic as the science of those “mental laws to which every sound thinker is bound to

conform.”95 After all, it is only a contingent fact about us that we can make errors and the

logical works written by beings for whom logical laws were in fact natural laws would

look the same as ours. Sigwart, while granting that logic is the ethics and not the physics

of thinking,96 nevertheless argues that taking some kinds of thinking and not others as

normative can only be justified psychologically – by noting when we experience the

“immediate consciousness of evident truth.”97 Mill also thinks that logical laws are

grounded in psychological facts: we infer the principle of contradiction, for instance,

from the introspectible fact that “Belief and Disbelief are two different mental states,

excluding one another.”98

For Lotze, the distinction between truth and “untruth” is absolutely fundamental

to logic but of no special concern to psychology. Thus psychology can tell us how we

93 Edmund Husserl, Logische Untersuchungen. Erster Teil: Prolegomena zur reinen Logik (Halle, 1900), §19. Mill, Examination, 359.

94 Mill, Logic, 12–13.95 Mansel, Prolegomena, 16.96 Sigwart, Logic, I 20 (Logik, I 22).97 Sigwart, Logic, I §3 (Logik, I §3).98 Mill, Logic, 277. But Mill appears to equivocate. He elsewhere suggests that the principle is

merely verbal after all: Mill, “Grote’s Aristotle,” reprinted in vol. 11 of Collected Works of John Stuart Mill, ed. J. M. Robson (Toronto: University of Toronto Press, 1978), 499–500. (Compare here Logic, 178.) Still elsewhere, he seems agnostic whether the principle is grounded in the innate constitution of our mind: Examination, 381.

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come to believe logical laws, but it cannot ground their truth.99 Frege argued in a similar

vein that psychology investigates how humans come to hold a logical law to be true but

has nothing to say about the law’s being true.100

Lotze thus supplemented the antipsychologistic arguments arising from the

normativity of logic with a separate line of argumentation based on the objectivity of the

domain of logic. For Lotze, the subjective act of thinking is distinct from the objective

content of thought. This content “presents itself as the same self-identical object to the

consciousness of others.”101 Logic, but not psychology, concerns itself with the objective

relations among the objective thought contents.102 Lotze was surely not the first

philosopher to insist on an act/object distinction. Herbart had earlier improved on the

ambiguous talk of “concepts” by distinguishing among the act of thinking (the

conceiving), the concept itself (that which is conceived), and the real things that fall under

the concept.103 But Lotze’s contribution was to use the act/object distinction to secure the

objectivity or sharability of thoughts. He interprets and defends Plato’s doctrine of Ideas

as affirming that no thinker creates or makes true the truths that he thinks.104

99 Lotze, Logic, §X; §332.100 Gottlob Frege, “Logic,” trans. Peter Lond and Roger White in Posthumous Writings, ed. Hans

Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Oxford: Basil Blackwell, 1979), 145. “Logik,” in Nachgelassene Schriften, ed. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Hamburg: Felix Meiner, 1969), 157. See also Frege, Foundations of Arithmetic, trans. J. L. Austin (Oxford: Blackwell, 1950), vi. Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl (Breslau, 1884), vi. (Identical paginations in German and English editions.)

101 Lotze, Logic, §345.102 Lotze, Logic, §332.103 Johann Friedrich Herbart, Lehrbuch zur Einleitung in die Philosophie, reprinted in Johann

Friedrich Herbart’s Sämmtliche Werke. Erster Band (Leipzig, 1850), §§34–5. (Original edition, 1813.) Independently, Mill severely criticized what he called “conceptualism” – a position common “from Descartes downwards, and especially from the era of Leibniz and Locke” – for confusing the act of judging with the thing judged. See Mill, Logic, 87–9.

104 Lotze, Logic, §313ff. On this nonmetaphysical reading of Plato, recognizing the sharability and judgment-independence of truth (§314) does not require hypostasizing the contents of thought or confusing the kind of reality they possess (which Lotze influentially called

25

Like Lotze, Frege associated the confusion of logic with psychology with erasing

the distinction between the objective and the subjective.105 Concepts are not something

subjective, such as an idea, because the same concept can be known intersubjectively.

Similarly, “We cannot regard thinking as a process which generates thoughts. . . . For do

we not say that the same thought is grasped by this person and by that person?”106 For

Lotze, intersubjective thoughts are somehow still products of acts of thinking.107 For

Frege, though, a thought exists independently of our thinking – it is “independent of our

thinking as such.”108 Indeed, for Frege any logic that describes the process of correct

thinking would be psychologistic: logic entirely concerns the most general truths

concerning the contents (not the acts) of thought.109

There are then multiple independent theses that one might call

“antipsychologistic.” One thesis asserts the independence of logic from psychology.

Another insists on the distinction between descriptive, psychological laws and normative,

logical laws. Another thesis denies that logical laws can be grounded in psychological

facts. Yet another thesis denies that logic concerns processes of thinking at all. Still

another thesis emphasizes the independence of the truth of thought-contents from acts of

holding-true. A stronger thesis maintains the objective existence of thought-contents.

Although Bernard Bolzano’s Theory of Science was published in 1837, it was

largely unread until the 1890s, when it was rediscovered by some of Brentano’s

“validity”) with the existence of things in space and time (§316).105 Frege, Foundations, x.106 Frege, Foundations, vii, §4. Frege, “Logic,” 137 (Nachgelassene, 148–9).107 Lotze, Logic, §345.108 Frege, Frege Reader, 206 (Grundgesetze, I xxiv). Frege, “Logic,” 133 (Nachgelassene, 144–

5).109 Frege, “Logic,” 146 (Nachgelassene, 158). Frege, Frege Reader, 202 (Grundgesetze, I xv).

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students.110 Bolzano emphasized more strongly than any thinker before Frege both that

the truth of thought-contents is independent of acts of holding-true and that there are

objective thought-contents.111 A “proposition in itself” is “any assertion that something is

or is not the case, regardless whether somebody has put it into words, and regardless even

whether or not it has been thought.”112 It differs both from spoken propositions (which are

speech acts) and from mental propositions insofar as the proposition in itself is the

content of these acts. Propositions and their parts are not real, and they neither exist nor

have being; they do not come into being and pass away.113

4. THE DOCTRINE OF TERMS

For the remainder of this article, we move from consideration of how the various

conceptions of logic evolved throughout the century to an overview of some of the

logical topics discussed most widely in the period. According to the tradition, a logic text

began with a section on terms, moved on to a section on those judgments or propositions

composed of terms, and ended with a section on inferences. Many of the works of the

century continued to follow this model, and we will follow suit here.

A fundamental debate among nineteenth-century logicians concerned what the

most basic elements of logic are. In the early modern period, Arnauld and Nicole had

110 Bernard Bolzano, Theory of Science, ed. and trans. Rolf George (Berkeley: University of California Press, 1972). The now-standard German edition is Bernard Bolzano, Wissenschaftslehre, ed. Jan Berg as vols. 11–14 of series 1 of Bernard-Bolzano-Gesamtausgabe, ed. Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromír Louzil, and Bob van Rootselaar (Stuttgart: Frommann, 1985–2000). Cited by paragraph numbers (§), common to English and German editions. As an example of Bolzano’s belated recognition, see Husserl, Logische Untersuchungen, 225–7.

111 Bolzano, Theory of Science, §48 (Wissenschaftslehre, 11.2: §48).112 Ibid., §19 (Wissenschaftslehre, 11.1:104).113 Ibid., §19 (Wissenschaftslehre, 11.1:105).

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transformed the doctrine of terms into the doctrine of ideas.114 Kant, having distinguished

intuitions from concepts, restricts the province of logic to conceptual representations.115

Mill, taking seriously that language is the chief instrument of thinking,116 begins with a

discussion of names, distinguishing between general and individual, categorematic and

syncategorematic, and connotative and nonconnotative (or denotative) names.117

For Mill, the attribute named by the predicate term in a proposition is affirmed or

denied of the object(s) named by the subject term.118 Mill opposes this view to the

common British theory that a proposition expresses that the class picked out by the

subject is included in the class picked out by the predicate.119 Though Mill thinks that

“there are as many actual classes (either of real or of imaginary things) as there are

general names,”120 he still insists that the use of predicate terms in affirming an attribute

of an object is more fundamental and makes possible the formation of classes. The debate

over whether a proposition is fundamentally the expression of a relation among classes or

a predication of an attribute overlapped with the debate over whether logic should

consider terms extensionally or intensionally.121 Logicians after Arnauld and Nicole

114 Antoine Arnauld and Pierre Nicole, Logic or the Art of Thinking, trans. Jill Vance Buroker (New York: Cambridge University Press, 1996).

115 Kant, Logic, §1. Subsequent German “formal” logicians followed Kant, as did the Kantian formal logicians in midcentury Britain. Thus, both Hamilton (Logic, I 13, 75, 131) and Mansel (Prolegomena, 22, 32) begin their works by distinguishing concepts from intuitions.

116 Mill, Logic, 19–20. Mill’s procedure draws on Whately, who claimed not to understand what a general idea could be if not a name; see Whately, Elements, 12–13, 37.

117 Many of these distinctions had been made earlier by Whately, and – indeed – by Scholastics. See Whately, Elements, 81.

118 Mill, Logic, 21, 97.119 Mill, Logic, 94. For an example of the kind of position Mill is attacking, see Whately,

Elements, 21, 26.120 Mill, Logic, 122.121 Mill’s view actually cuts across the traditional distinction: he thinks that the intension of the

predicate name is an attribute of the object picked out by the subject name. Traditional intensionalists thought that the intension of the predicate term is contained in the intension of the subject term; traditional extensionalists thought that the proposition asserts that the extension of the predicate term includes the extension of the subject term.

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distinguished between the intension and the extension of a term. The intension (or

content) comprises those concepts it contains. The extension of a term is the things

contained under it – either its subspecies122 or the objects falling under it. Hamilton

thought that logic could consider judgments both as the inclusion of the extension of the

subject concept in the extension of the predicate concept and as the inclusion of the

predicate concept in the intension of the subject concept.123 But, as Mansel rightly

objected, if a judgment is synthetic, the subject class is contained in the predicate class

without the predicate’s being contained in the content of the subject.124

Boole self-consciously constructed his symbolic logic entirely extensionally.125

Though Jevons insisted that the intensional interpretation of terms is logically

fundamental,126 the extensional interpretation won out. Venn summarized the common

view when he said that the task of symbolic logic is to find the solution to the following

problem: given any number of propositions of various types and containing any number

of class terms, find the relations of inclusion or inclusion among each class to the rest.127

122 See, e.g., Kant, Logic, §7–9. In the critical period, when Kant more thoroughly distinguished relations among concepts from relations among objects, Kant began to talk also of objects contained under concepts (e.g., A137/B176).

123 Hamilton, Lectures, I 231–2. This position was also defended by Hamilton’s student William Thomson, An Outline of the Necessary Laws of Thought, 2nd ed. (London, 1849), 189.

124 Henry Longueville Mansel, “Recent Extensions of Formal Logic,” reprinted in Letters, Lectures, and Reviews, ed. Henry W. Chandler (London, 1873), 71. (Original edition, 1851.)

125 “What renders logic possible is the existence in our mind of general notions – our ability . . . from any conceivable collection of objects to separate by a mental act those which belong to the given class and to contemplate them apart from the rest.” Boole, Laws of Thought, 4.

126 Jevons, “Pure Logic,” §1-5, 11, 17. 127 Venn, Symbolic Logic, xx.

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Frege sharply distinguished singular terms from predicates128 and later even the

referents of proper names from the referents of predicates.129 In the traditional logic,

“Socrates is mortal” and “Humans are mortal” were treated in the same way. Thus, for

Kant, both “Socrates” and “Human” express concepts; every subjudgmental component

of a judgment is a concept.130 Frege therefore also departed from the traditional logic in

distinguishing the subordination of one concept to another from the subsumption of an

object under a concept. This distinction was not made in the Boolean tradition; for Boole,

variables always refer to classes, which are thought of as wholes composed of parts.131

Thus a class being a union of other classes is not distinguished from a class being

composed of its elements.

The traditional doctrine of concepts included a discussion of how concepts are

formed. On the traditional abstractionist model, concepts are formed by noticing

similarities or differences among particulars and abstracting the concept, as the common

element.132 This model came under severe criticism from multiple directions throughout

the century. On the traditional “bottom-up” view, the concept F is formed from

128 Gottlob Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle, 1879), §9, trans. Michael Beaney in The Frege Reader. Citations are by paragraph numbers (§), common to the German and English editions, or by page numbers from the original German edition, which are reproduced in the margins of The Frege Reader.

129 Frege, “Funktion und Begriff,” in Kleine Schriften, ed. Igancio Angelelli (Darmstadt: Wissenschaftliche Buchgesellschaft, 1967), 125–42, trans. Peter Geach and Brian McGuiness as “Function and Concept” in Collected Papers on Mathematics, Logic, and Philosophy, ed. Brian McGuiness (New York: Basil Blackwell, 1984).

130 The Kantian distinction between concepts and intuitions is thus not parallel to Frege’s concept/object distinction.

131 Frege points out that the Booleans failed to make this distinction in his paper “Boole’s Logical Calculus and the Concept-script,” trans. Peter Lond and Roger White in Posthumous Writings, ed. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Oxford: Basil Blackwell, 1979), 18. Nachgelassene Schriften, ed. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach (Hamburg: Felix Meiner, 1969), 19–20. This essay was written in 1880–1. Unknown to Frege, Bolzano had clearly drawn the distinction forty years earlier – see Bolzano, Theory of Science, §66.2, §95 (Wissenschaftslehre, 11.2:105–6, 11.3:38–9).

132 See, e.g., Kant, Logic, §6.

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antecedent representations of particular Fs, and judgments containing F are formed

subsequent to the acquisition of the concept. For Frege, the order of priority is reversed:

“As opposed to [Boole], I start out with judgments and their contents, and not from

concepts. . . . I only allow the formation of concepts to proceed from judgments.”133 The

concepts 4th power and 4th root of 16 are formed, not by abstraction, but by starting with

the judgment “24 = 16” and replacing in its linguistic expression one or more singular

terms by variables. These variables can then be bound by the sign for generality to form

the quantified relational expressions that give Frege’s new logic its great expressive

power. Since the same judgment can be decomposed in different ways, a thinker can form

the judgment “24 = 16” without noting all of the ways in which the judgment can be

decomposed. This in turn explains how the new logic can be epistemically ampliative.134

Kant had earlier asserted his own kind of “priority thesis” (A68–9/B93–4), and

the historical connection between Kant and Frege’s priority principles is a complicated

one that well illustrates how philosophical and technical questions became intertwined

during the century. For Kant, concepts are essentially predicates of possible judgments,

because only a judgment is subject to truth or falsity (B141–2), and thus it is only in

virtue of being a predicate in a judgment that concepts relate to objects (Ak 4:475).

Though Kant never explicitly turned this thesis against the theory of concept formation

by abstraction, Hegel rightly noted that implicit within Kantian philosophy is a theory

opposed to abstractionism.135 For Kant, just as concepts are related to objects because

they can be combined in judgments, intuitions are related to objects because the manifold

contained in an intuition is combined according to a certain rule provided by a concept 133 Frege, “Boole’s Calculus,” 16–17 (Nachgelassene, 17–19).134 Ibid., 33–4 (Nachgelassene, 36–8). Frege, Foundations, §88.135 Hegel, Science of Logic, 589 (Werke 12:22).

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(A150). Thus, the theory of concept formation by abstraction cannot be true in general:

the very representation of particulars in intuition already requires the possession of a

concept. This Kantian-inspired Hegelian argument was directed against Hamilton and

Mansel in T. H. Green’s 1874–5 “Lectures on the Formal Logicians” and against Boolean

logic by Robert Adamson.136 (Indeed, these works imported post-Kantian reflections on

logic into Britain, setting the stage for the idealist logics that gained prominence later in

the century.)

Hegel added a second influential attack on abstractionism. The procedure of

comparing, reflecting, and abstracting to form common concepts does not have the

resources to discriminate between essential marks and any randomly selected common

feature.137 Trendelenburg took this argument one step further and claimed that it was not

just the theory of concept formation, but also the structure of concepts in the traditional

logic that was preventing it from picking out explanatory concepts. A concept should

“contain the ground of the things falling under it.”138 Thus, the higher concept is to

provide the “law” for the lower concept, and – pace Drobisch139 – a compound concept

cannot just be a sum of marks whose structure could be represented using algebraic signs:

human is not simply animal + rational.140

136 Thomas Hill Green, “Lectures on Formal Logicians,” in Works of Thomas Hill Green, 3 vols., ed. R. L. Nettleship (London, 1886), 165, 171. Adamson, History, 117, 122.

137 Hegel, Science of Logic, 588 (Werke 12:21). Again, this argument was imported to Britain much later in the century. See Green, “Lectures,” 193; Adamson, History, 133–4.

138 Trendelenburg, Untersuchungen, I 18–19.139 Moritz Wilhelm Drobisch, Neue Darstellung der Logik, 2nd ed. (Leipzig, 1851), ix, §18.

Drobisch defended the traditional theory of concept formation against Trendelenburg’s attack, occasioning an exchange that continued through the various editions of their works.

140 Trendelenburg, Untersuchungen, I 20. Trendelenburg, “Über Leibnizens Entwurf einer allgemeinen Characteristik,” reprinted in Historische Beiträge zur Philosophie, vol. 3 (Berlin, 1867), 24. (Original edition, 1856.) Trendelenburg thought that his position was Aristotelian. For Aristotle, there is an important metaphysical distinction between a genus and differentia. Yet this distinction is erased when the species concept is represented, using a commutative operator such as addition, as species = genus + differentia.

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Lotze extended and modified Trendelenburg’s idea. For him, the “organic

bond”141 among the component concepts in a compound concept can be modeled

“functionally”: the content of the whole concept is some nontrivial function of the

content of the component concepts.142 Concepts formed by interrelating component

universals such as interdependent variables in a function can be explanatory, then,

because the dependence of one thing on another is modeled by the functional dependence

of component concepts on one another. Lotze thinks that there are in mathematics kinds

of inferences more sophisticated than syllogisms (§106ff.) and that it is only in these

mathematical inferences that the functional interdependence of concepts is exploited

(§120). Boolean logic, however, treats concepts as sums and so misses the functionally

compound concepts characteristic of mathematics.143

Frege is thus drawing on a long tradition – springing ultimately from Kant but

with significant additions along the way – when he argues that the abstractionist theory of

concept formation cannot account for fruitful or explanatory concepts because the

“organic” interconnection among component marks in mathematical concepts is not

respected when concepts are viewed as sums of marks.144 But Frege was the first to think

of sentences as analyzable using the function/argument model145 and the first to

appreciate the revolutionary potential of the possibility of multiple decompositionality.

Sigwart gave a still more radical objection to abstractionism. He argued that in

order to abstract a concept from a set of representations, we would need some principle

for grouping together just this set, and in order to abstract the concept F as a common 141 This is Trendelenburg’s Aristotelian language: see Trendelenburg, Untersuchungen, I 21.142 Lotze, Logic, §28.143 Lotze, Logic, I 277ff. (Lotze, Logik, 256ff.).144 Frege, Foundations, §88.145 Frege, Begriffsschrift, vii.

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element in the set, we would need already to see them as Fs. The abstractionist theory is

thus circular and presupposes that ability to make judgments containing the concept.146

5. JUDGMENTS AND INFERENCES

The most common objection to Kant’s table of judgments was that he lacked a principle

for determining that judgment takes just these forms. Hegel thought of judging as

predicating a reflected concept of a being given in sensibility – and so as a relation of

thought to being. The truth of a judgment would be the identity of thought and being, of

the subject and predicate. Hegel thus tried to explain the completeness of the table of

judgments by showing how every form of judgment in which the subject and predicate

fail to be completely identical resolves itself into a new form.147 For Hegel, then, logic

acquires systematicity not through reducing the various forms of judgment to one

another, but by deriving one from another. Other logicians, including those who rejected

Hegel’s metaphysics, followed Hegel in trying to derive the various forms from one

another.148

British logicians, on the other hand, tended to underwrite the systematicity of

logic by reducing the various forms of judgment to one common form. Unlike Kant,149

Whately reduced disjunctive propositions to hypotheticals in the standard way, and he

reduced the hypothetical judgment “If A is B, then X is Y,” to “The case of A being B is a

146 Sigwart, Logic, §40.5. Sigwart thus initiated the widely repeated practice of inverting the traditional organization of logic texts, beginning with a discussion of judgments instead of concepts. Lotze himself thought that Sigwart had gone too far in his objections to abstractionism: Lotze, Logic, §8.

147 Hegel, Science of Logic, 625–6, 630 (Werke 54–5, 59).148 E.g., Lotze, Logic, I 59 (Lotze, Logik, 70).149 A73–4/B98–99. Later German logicians tended to follow Kant. See, for example, Fries, Logik,

§32.

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case of X being Y.”150 All judgments become categorical, all reasoning becomes

syllogizing, and every principle of inference reducible to Aristotle’s dictum de omni et

nullo. Mansel was more explicit in reading hypothetical judgments temporally: he

interprets “If Caius is disengaged, he is writing poetry” as “All times when Caius is

disengaged are times when he is writing poetry.”151 Mill endorses Whately’s procedure,

interpreting “If p then q” as “The proposition q is a legitimate inference from the

proposition p.”152 Mill, of course, thought that syllogistic reasoning is really grounded in

induction. But in feeling the need to identify one “universal type of the reasoning

process,” Mill was at one with Whately, Hamilton, and Mansel, each of whom tried to

reduce induction to syllogisms.153

Boole notoriously argues that one and the same logical equation can be

interpreted either as a statement about classes of things or as a “secondary proposition” –

a proposition about other propositions.154 Let “x” represent the class of times in which the

proposition X is true. Echoing similar proposals by his contemporaries, Boole expresses

“If Y then X” as “y = vx”: “The time in which Y is true is an indefinite portion of the time

in which X is true.”155 As Frege pointed out, making the calculus of classes and the

calculus of propositions two distinct interpretations of the same equations prevents Boole

from analyzing the same sentence using quantifiers and sentential operators

simultaneously.156 Frege’s Begriffsschrift gave an axiomatization of truth functional

150 Whately, Elements, 71, 74–5.151 Mansel, Prolegomena, 197.152 Mill, Logic, 83–4.153 Mill, Logic, 202. Whately, Elements, 153. Hamilton, “Recent Publications,” 162. Mansel,

Prolegomena, 191.154 Boole, Laws of Thought, 160.155 Ibid, 170.156 Frege, “Boole’s Calculus,” 14–15 (Nachgelassene, 15–16).

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propositional logic that depends on neither the notion of time nor a calculus of classes. In

this, Frege was anticipated by Hugh MacColl.157 Independently, Peirce axiomatized two-

valued truth-functional logic, clearly acknowledging that the material conditional, having

no counterfactual meaning, differs from the use of “if” in natural language.158

Among the most discussed and most controversial innovations of the century

were Hamilton’s and DeMorgan’s theories of judgments with quantified predicates. In

the traditional logic, the quantifier terms “all” and “some” are only applied to the subject

term, and so “All As are Bs” does not distinguish between the case where the As are a

proper subset of the Bs (in Hamilton’s language: “All As are some Bs”) and the case

where the As are coextensive with the Bs (“All As are all Bs”). Now, a distinctive

preoccupation of logicians in the modern period was to arrive at systematic methods that

would eliminate the need to memorize brute facts about which of the 256 possible cases

of the syllogism were valid. A common method was to reduce all syllogisms to the first

figure by converting, for example, “All As are Bs” to “Some Bs are As.”159 This required

students to memorize which judgments were subject to which kinds of conversions.

Quantifying the predicate, however, eliminates the distinctions among syllogistic figures

and all of the special rules of conversion: all conversion becomes simple – “All A is some

B” is equivalent to “Some B is all A.”160

157 Hugh MacColl, “The Calculus of Equivalent Statements,” Proceedings of the London Mathematical Society 9 (1877): 9–20, 177–86.

158 Charles Sanders Peirce, “On the Algebra of Logic: A Contribution to the Philosophy of Notation,” reprinted in Collected Papers of Charles Sanders Peirce, vol. 3, Exact Logic, ed. Charles Hartshorne and Paul Weiss (Cambridge: Harvard University Press, 1933), 218–19. (Original edition, 1885.)

159 See, e.g., Whately, Elements, 61. For an earlier example, see Immanuel Kant, The False Subtlety of the Four Syllogistic Figures, in Theoretical Philosophy: 1755–1770, ed. And trans. D. Walford and R. Meerbote (Cambridge: Cambridge University Press, 1992).

160 Hamilton, Lectures, II 273.

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Given the simplification allowed by predicate quantification, Hamilton thought he

could reduce all of the syllogistic rules to one general canon.161 De Morgan rightly argued

that some of Hamilton’s new propositional forms are semantically obscure,162 and he

independently gave his own system and notation for quantified predicates. (Hamilton

then initiated a messy dispute over priority and plagiarism with De Morgan.) In De

Morgan’s notation, there are symbols for the quantity of terms (parentheses), for the

negation of the copula (a dot), and – what was new in De Morgan – for the contrary or

complement of a class (lowercase letters). “All As are (some) Bs” is “A))B.” DeMorgan

gave rules for the interaction of quantification, class contraries, and copula negation.163

The validity of syllogisms is demonstrated very easily, by a simple erasure rule: Barbara

is “A))B; B))C; and so A))C.”164

In traditional logic, negation was always attached to the copula “is,” and it did not

make sense to talk – as De Morgan did – of a negated or contrary term.165 Further

departing from tradition, Frege thought of negation as applied to whole sentences and not

just to the copula.166 (Boole, of course, had already effectively introduced negation as a

sentential operator: he expressed the negation of X as “1 - x.”)167 The introduction of

contrary terms led De Morgan to restrict possible classes to a background “universe

161 Hamilton, Lectures, II 290–1.162 De Morgan, “Logic,” 257–8.163 Augustus De Morgan, Syllabus of a Proposed System of Logic, reprinted in On the Syllogism,

157ff. (Original edition, 1860.)164 De Morgan, “Syllogism II,” 31.165 As Mansel correctly pointed out: Mansel, “Recent Extensions,” 64.166 Frege, Begriffsschrift, §7.167 Boole, Laws of Thought, 168.

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under consideration.” If the universe is specified (say, as living humans), then if A is the

class of Britons, a is the class of humans who are not Britons.168

De Morgan’s work on relations led him to distinguish between the relation

between two terms and the assertion of that relation, thus separating what was often

confused in traditional discussions of the function of the copula.169 But lacking a sign for

propositional negation, De Morgan did not explicitly distinguish between negation as a

sentential operator and an agent’s denial of a sentence – a mistake not made by Frege.170

De Morgan’s logic of relations was one of many examples of a logical innovation

hampered by its adherence to the traditional subject-copula-predicate form. Although

Bolzano was quite clear about the expressive limitations of the traditional logic in other

respects, he nevertheless forced all propositions into the triadic form “subject has

predicate.”171 In a sense, the decisive move against the subject-copula-predicate form

was taken by Boole, since an equation can contain an indefinite number of variables, and

there is no sense in asking which term is the subject and which is the predicate. Frege

went beyond Boole in explicitly recognizing the significance of his break with the

subject/predicate analysis of sentences.172

De Morgan required all terms in his system (and their contraries) to be

nonempty.173 With this requirement, the following nontraditional syllogism turns out

168 See De Morgan, Formal Logic, 37. Boole adopted De Morgan’s idea, renaming it a “universe of discourse” (Laws of Thought, 42).

169 De Morgan, “Syllogism IV,” 215. Compare Mill, Logic, 87.170 Frege, Begriffsschrift, §2.171 Bolzano, Theory of Science, §127 (Wissenschaftslehre, 12.1:70–1).172 Frege, Begriffsschrift, vii. Frege’s break with the subject-predicate analysis of sentences also

made irrelevant the development of systems of quantified predicates among British logicians.173 De Morgan, Formal Logic, 127.

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valid: “All Xs are Ys; all Zs are Ys; therefore, some things are neither Xs nor Zs.”174 Boole,

on the other hand, did not assume that the class symbols in his symbolism be

nonempty.175 The debate over the permissibility of terms with empty extensions

dovetailed with long-standing debates over the traditional doctrine that universal

affirmative judgments imply particular affirmative judgments.176 Herbart denied that the

subject term in a judgment “A is B” must exist, since (for example) we can judge that the

square circle is impossible; Fries argued that “Some griffins are birds” is false, even

though “All griffins are birds” is true.177 Although Boole himself followed the tradition,178

later Booleans tended to follow Herbart and Fries.179 Independently Brentano, in keeping

with the “reformed logic” made possible largely by his existential theory of judgment,

read the universal affirmative as “There is no A that is non-B” and denied that it implied

the particular affirmative.180

174 De Morgan, “Syllogism II,” 43. This syllogism is valid because – by De Morgan’s requirement that all terms and their contraries be nonempty – there must be some non-Ys, and from the two premises we can infer that the non-Ys cannot be Xs or Zs. So some things (namely, the non-Ys) are neither Xs nor Zs.

175 Boole, Laws of Thought, 28.176 For an example of the traditional view, see Whately, Elements, 46–7. Bolzano also defended

the tradition: Theory of Science, §225 (Wissenschaftslehre, 12.3:57–9).177 Herbart, Lehrbuch, §53. Fries, Logik, 123.178 Boole, Laws of Thought, 61. He represented universal affirmatives as “x = vy” and particular

affirmatives as “vx = vy” with v the symbol for some indefinite selection; given the assumption that vx is always nonzero (Laws of Thought, 61) and that v2 = v, the inference holds (Laws of Thought, 229). In general, if a logician allowed for terms with empty extensions, then the inference from universal affirmative to particular affirmative would fail. But Boole illustrates that this holds only in general.

179 Charles Sanders Peirce, “On the Algebra of Logic,” reprinted in Collected Papers of Charles Sanders Peirce, Vol. 3, Exact Logic, ed. Charles Hartshorne and Paul Weiss (Cambridge: Harvard University Press, 1933), 114. (Original edition, 1880.) Venn, Symbolic Logic, 141ff.

180 Franz Brentano, Psychology from an Empirical Standpoint, trans. A. C. Rancurello, D. B. Terrell, and L. McAlister, ed. Peter Simons, 2nd ed. (London: Routledge, 1995), 230; Psychologie vom empirischen Standpunkt, 2nd ed. (Leipzig: Felix Meiner, 1924), II 77. (Original edition, 1874.)

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Whately argued that syllogistic was grounded in one principle only, Aristotle’s

dictum de omni et nullo: “What is predicated, either affirmatively or negatively, of a term

distributed, may be predicated in like manner (affirmatively or negatively) of any thing

contained under that term.”181 Hamilton thought that the dictum was derivable from the

more fundamental law: “The part of the part is the part of the whole.”182 Mill rejected the

dictum and identified two principles of the syllogism: “Things which coexist with the

same thing, coexist with one another,” and “A thing which coexists with another thing,

with which other a third thing does not coexist, is not coexistent with that third thing.”

These principles are laws about facts, not ideas, and (he seems to suggest) they are

grounded in experience.183 Mansel argued that the dictum could be derived from the more

fundamental principles of identity and contradiction, a position taken earlier by

Twesten.184 De Morgan argued that the validity of syllogisms depends on the transitivity

and commutativity of the copula. He argues against Mansel that these two properties

cannot be derived from the principles of contradiction and identity (which gives

reflexivity, not commutativity or transitivity).185

By 1860, De Morgan considered syllogistic to be just one material instantiation of

the most general form of reasoning: “A is an L of B; B is an M of C; therefore, A is an L

of an M of C.” Nevertheless, De Morgan thought that syllogistic needed completing, not

discarding: his logic of relations is the “higher atmosphere of the syllogism.”186 Hamilton

intended his system of quantified predicates to “complete and simplify the old; – to place

181 Whately, Elements, 31.182 Hamilton, Lectures, I 144–5; compare Kant, Logic, §63.183 Mill, Logic, 178.184 Mansel, Prolegomena, 189. August Twesten, Die Logik, insbesondere die Analytik

(Schleswig, 1825), §6.185 De Morgan, Formal Logic, 50ff.; De Morgan, “Syllogism IV,” 214.186 De Morgan, “Syllogism IV,” 241.

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the keystone in the Aristotelic arch.”187 Similarly, Venn later argued that Boole’s

symbolic logic generalizes and develops the common logic, and he advocates retaining

the old syllogistic logic in the classroom.188 (This attitude contrasts sharply with the

contempt for syllogistic shown by German logicians in the generation of Schleiermacher

and Hegel.)

Perhaps the deepest and most innovative contribution to the theory of inference

was Bolzano’s theory of deducibility, based on the method of idea variation that he

introduced in his Theory of Science. The propositions C1, …, Cn are deducible from P1,

…, Pm with respect to some idea i if every substitution of an idea j for i that makes P1, …,

Pm true also makes C1, …, Cn true.189 If we restrict our attention to those inferences where

the conclusions are deducible with respect to all logical ideas,190 we isolate a class of

“formal” inferences. Bolzano is thus able to pick out all of the logically correct inferences

in a fundamentally different way from his contemporaries: he does not try to reduce all

possible inferences to one general form, and he does not need to ground the validity of

deductions in an overarching principle, like Aristotle’s dictum or the principle of identity.

Bolzano admits, however, that he has no exhaustive or systematic list of logical ideas –

so his idea is not fully worked out.191

6. LOGIC, LANGUAGE, AND MATHEMATICS

Debate in Germany over the necessity or possibility of a new logical symbolism centered

around Leibniz’s idea of a “universal characteristic,” which was discussed in a widely

187 Hamilton, Lectures, II 251.188 Venn, Symbolic Logic, xxvii.189 Bolzano, Theory of Science, §155 (Wissenschaftslehre, 12.1:169–86).190 Bolzano, Theory of Science, §223 (Wissenschaftslehre, 12.3:47–9).191 Bolzano, Theory of Science, §148.2 (Wissenschaftslehre, 12.1:141).

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read paper by Trendelenburg. As Trendelenburg describes the project, Leibniz wanted a

language in which, first, the parts of the symbols for a compound concept would be

symbols for the parts of the concept itself, and, second, the truth or falsity of any

judgment could be determined by calculating.192 To develop such a language would

require first isolating all of the simple concepts or categories (20). Trendelenburg thought

the project was impossible. First, it is not possible to isolate the fundamental concepts of

a science before the science is complete, and so the language could not be a tool of

scientific discovery (25). Second, Leibniz’s characterization of the project presupposes

that all concepts can be analyzed as sums of simple concepts, and all reasoning amounts

to determining whether one concept is contained in another. Thus, Leibniz’s project is

subject to all of the objections, posed by Trendelenburg and others, to the abstractionist

theory of concept formation and the theory of concepts as sums of marks (24).

The title of Frege’s 1879 book – Begriffsschrift or “Concept-script” – is taken

from Trendelenburg’s essay.193 In his 1880 review, Schröder argues that Frege’s title does

not correspond to the content of the book.194 A “Begriffsschrift,” or universal

characteristic, would require a complete analysis of concepts into basic concepts or

“categories” and a proof that the content of every concept can be formed from these

categories by a small number of operations. To Schröder, Frege’s project is closer to a

related Leibnizian project, the development of a calculus ratiocinator, a symbolic

calculus for carrying out deductive inferences but not for expressing content. In reply,

192 Trendelenburg, “Leibnizens Entwurf,” 6, 18.193 See Trendelenburg, “Leibnizens Entwurf,” 4.194 Ernst Schröder, “Review of Begriffsschrift, by Gottlob Frege,” trans. Terrell Ward Bynum in

Conceptual Notation and Related Articles (Oxford: Oxford University Press, 1972). (Original edition: “Rezension von Gottlob Frege, Begriffsschrift,” Zeitschrift für Mathematik und Physik 25 [1880]: 81–94.)

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Frege argued that his begriffsschrift does differ from Boolean logic in aiming to be both a

calculus ratiocinator and a universal characteristic.195 To carry out his logicist project,

Frege needs to isolate the axioms of arithmetic, show that these axioms are logical truths

and that every concept and object referred to in these axioms is logical, define

arithmetical terms, and finally derive the theorems of arithmetic from these axioms and

definitions. Since these proofs need to be fully explicit and ordinary language is

unacceptably imprecise, it is clear that a logically improved language is needed for

expressing the content of arithmetic.196 Moreover, in strongly rejecting the traditional

view that concepts are sums of marks and that all inferring is syllogizing, Frege was

answering the objections to Leibniz’s project earlier articulated by Trendelenburg.

Schröder’s 1877 Der Operationskreis des Logikkalküls opens by arguing that

Boole had fulfilled Leibniz’s dream of a logical calculus.197 Later he argued explicitly

that his algebra of logic was a necessary and significant step in the development of a

universal characteristic or “pasigraphy.”198 Venn, on the other hand, argued that the

symbolic logic developed since Boole differs from Leibniz’s universal characteristic “as

language should and does differ from logic.” In symbolic logic, each symbol is a variable

standing for any class whatsoever; in a universal characteristic, the symbols refer to

definite classes.199

195 Frege, “Boole’s Calculus,” 12 (Nachgelassene, 13).196 Gottlob Frege, “On the Scientific Justification of a Conceptual Notation,” trans. Terrell Ward

Bynum in Conceptual Notation and Related Articles (Oxford: Oxford University Press, 1972), 85. (Original edition: “Über die wissenschaftliche Berechtigung einer Begriffsschrift,” Zeitschrift für Philosophie und philosophische Kritik 81 [1882]: 50–1.)

197 Schröder, Operationskreis, iii.198 Ernst Schröder, Vorlesungen über die Algebra der Logik, vol. 1 (Leipzig, 1890), 95. Ernst

Schröder, “On Pasigraphy. Its Present State and the Pasigraphic Movement in Italy,” Monist 9 (1898): 44–62.

199 Venn, Symbolic Logic, 109.

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Frege’s thesis that arithmetic is a branch of logic200 was but one contribution in

the long debate about the relation between logic and mathematics. An old debate was

whether mathematical proofs – specifically, geometrical proofs – could be cast in

syllogistic form. Euler had thought so; Schleiermacher did not.201 Thomas Reid had

argued that an inference involving a judgment with three terms – such as an instance of

the transitivity of equality – could not be captured in syllogisms. Hamilton, in his 1846

note in his edition of Reid’s works, argues that one can express transitivity of equality

syllogistically as “What are equal to the same are equal to each other; A and C are equal

to the same (B); therefore, A and C are equal to each other.”202 As De Morgan rightly

noted, this syllogism does not reduce the transitivity of equality; it presupposes it.203

Both De Morgan and Boole wanted to make logic symbolic, in a way modeled on

mathematics. Mansel accused both Boole and De Morgan of treating logic as an

application of mathematics.204 This is a confusion, he contended, because logic is formal

and mathematics is material. Boole, unlike De Morgan, took from algebra specific

symbols, laws, and methods. Nevertheless, Boole argued that even though logic’s

“ultimate forms and processes are mathematical,” it is only established a posteriori that

200 Frege, Grundgesetze, 1.201 Leonard Euler, Letters of Euler on Different Topics in Natural Philosophy, Addressed to a

German Princess, vol. 1, trans. David Brewster (New York, 1833), 354. (Original edition, 1768–72.) Schleiermacher denied that syllogisms are sufficient for geometrical proofs, since the essential element is the drawing of the additional lines in the diagram: Dialektik, 287.

202 Thomas Reid, Essays on the Active Powers of the Human Mind, reprinted in The Works of Thomas Reid, D.D. Now Fully Collected, with Selections from His Unpublished Letters, with preface, notes, and supplementary dissertations by William Hamilton (Edinburgh, 1846), 702. (Reid’s Essays first appeared in 1788.) Hamilton’s argument appears as an editorial footnote on the same page.

203 De Morgan, “On the Syllogism II,” 67.204 Mansel, “Recent Extensions,” 47. Mansel was directing his argument against Boole’s Laws of

Thought and De Morgan’s Formal Logic.

44

the algebra of logic can be interpreted indifferently as an algebra of classes or

propositions, or again as an algebra of the quantities 0 and 1.205

Jevons accused Boole of, in essence, beginning with self-evident logical notions,

transforming them into a symbolism analogous to the algebra of the magnitudes 0 and 1,

manipulating the equations as if they were about quantities, and then interpreting them as

logical inferences – with no justification save the fact that they seem to work out in the

end.206 But this process gets the dependency backward: logic, being purely intensional (or

qualitative), is presupposed by the science of number (or quantity), since numbers are

composed of qualitatively identical but logically distinct units.207 For Venn, mathematics

and symbolic logic are best thought of as two branches of one language of symbols,

characterized by a few combinatorial laws. It would be acceptable to think of logic as a

branch of mathematics, as long as one understands mathematics – as Boole did – to be

“the science of the laws and combinations of symbols.”208

Lotze severely criticized Boole for justifying his method on the basis of “rash and

misty analogy drawn from the province of mathematics.”209 With respect to the relation

between the two disciplines, Lotze emphasized that “all calculation is a kind of thought,

that the fundamental concepts and principles of mathematics have their systematic place

in logic.”210 Though some commentators have seen this claim as a forerunner of Frege’s

logicism,211 Lotze means by this something more modest, and yet still very significant.

205 Boole, Laws of Thought, 12; 37.206 Jevons, “Pure Logic,” §202.207 Jevons, “Pure Logic,” §6; §§185–6.208 Venn, Symbolic Logic, xvi–ii.209 Lotze, Logic, I 277–98, esp. 278 (Logik, 256–69, esp. 256).210 Lotze, “Logic,” §18.211 Hans Sluga, Gottlob Frege (Boston: Routledge & Kegan Paul, 1980), 57. Sluga, “Frege: The

Early Years,” in Philosophy in History: Essays in the Historiography of Philosophy, ed. Richard Rorty, Jerome B. Schneewind, and Quentin Skinner (New York: Cambridge

45

Lotze is advocating that logicians analyze the distinctive kinds of conceptual structures

and inferences found in mathematics; such an analysis shows, Lotze thinks, that

mathematics outstrips the expressive capacity of syllogistic. Lotze identified three kinds

of mathematical inferences irreducible to syllogisms: “inference by substitution,”

“inference by proportion,” and “inference from constitutive equations.”212

University Press, 1984), 343–4. Gottfried Gabriel has argued for a similar conclusion; see his “Objektivität, Logik und Erkenntnistheorie bei Lotze und Frege,” in Logik. Drittes Buch. Vom Erkennen, by Hermann Lotze, ed. Gottfired Gabriel (Hamburg: Felix Meiner, 1989), ix–xxxvi.

212 Lotze, Logic, §§105–19.

46

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